A new report from the anti-immigrant Center for Immigration Studies purports to argue that immigration reformers (along with advocates) have hoodwinked the country into believing that there is a shortage of workers in the Science, Technology, Engineering, and Math (or STEM) fields, as a way to increase immigration to the U.S. CIS claims that there are double as many STEM degree holders as STEM jobs, and that wage growth in the STEM fields has been relatively stagnant. Adding more immigrants, according to CIS “seem[s] entirely divorced from what is actually going on in the U.S. labor market.”

First off, as even CIS acknowledges, the problem in the STEM fields is not a lack of native-born graduates with STEM training, it is that those graduates often do not stay in STEM fields. Georgetown’s Center on Education and the Workforce, for example, has found that many STEM workers take jobs in “a variety of occupations ranging from architecture to business and finance to medicine,” in no small part because many of these professions simply pay more than the STEM fields. Higher pay leads many of these workers to take their training elsewhere—think of the “quants” that went into investment banking prior to the financial crisis. Women, as the National Science Foundation has pointed out, also face a number of barriers to staying and advancing within the STEM fields, including discrimination and the demands of family.

Second, as Alex Nowrasteh of the Cato Institute points out, CIS attempts to show a lack of STEM shortage by arguing that wages for STEM workers have only grown slightly over the past decade. But CIS fails to compare these wages to the wages for all occupations. It turns out that over the past decade (2001-2012), real wages for all workers actually fell by close to 1 percent, while wages for STEM workers rose by over 2 percent.

Third, not all STEM jobs are equal: a job that requires a bachelor’s degree in computer science and a doctoral degree in bioengineering, for example, will require a very different skill set. As Change the Equation has argued, job postings in electrical engineering outnumber the unemployed in these categories by more than 1.3 to 1, while STEM job advertisements in healthcare outnumber the unemployed by more than 3 to 1. Arguing, as CIS attempts, that because certain fields have fewer job openings than others means that we have to restrict all STEM immigration, is throwing the baby out with the bathwater.

The bottom line: we may be training enough STEM workers in the U.S., but until we actually find a way to retain workers in the STEM fields (by, for example, increasing access to paid leave, which studies have shown makes women less likely to leave the labor market after having a child), our nation will have a mismatch between the skills that its businesses need to operate and innovate, and the available workers. We can and should do more to train and retrain STEM workers domestically. But make no mistake: immigrants play a vital role in the STEM fields.

Philip E. Wolgin is a Senior Policy Analyst for Immigration at the Center for American Progress